3.360 \(\int \frac{x \left (d+e x^2\right )^{3/2}}{a+b x^2+c x^4} \, dx\)

Optimal. Leaf size=327 \[ -\frac{\left (-2 c e \left (-d \sqrt{b^2-4 a c}+a e+b d\right )+b e^2 \left (b-\sqrt{b^2-4 a c}\right )+2 c^2 d^2\right ) \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{c} \sqrt{d+e x^2}}{\sqrt{2 c d-e \left (b-\sqrt{b^2-4 a c}\right )}}\right )}{\sqrt{2} c^{3/2} \sqrt{b^2-4 a c} \sqrt{2 c d-e \left (b-\sqrt{b^2-4 a c}\right )}}+\frac{\left (-2 c e \left (d \sqrt{b^2-4 a c}+a e+b d\right )+b e^2 \left (\sqrt{b^2-4 a c}+b\right )+2 c^2 d^2\right ) \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{c} \sqrt{d+e x^2}}{\sqrt{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}}\right )}{\sqrt{2} c^{3/2} \sqrt{b^2-4 a c} \sqrt{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}}+\frac{e \sqrt{d+e x^2}}{c} \]

[Out]

(e*Sqrt[d + e*x^2])/c - ((2*c^2*d^2 + b*(b - Sqrt[b^2 - 4*a*c])*e^2 - 2*c*e*(b*d
- Sqrt[b^2 - 4*a*c]*d + a*e))*ArcTanh[(Sqrt[2]*Sqrt[c]*Sqrt[d + e*x^2])/Sqrt[2*
c*d - (b - Sqrt[b^2 - 4*a*c])*e]])/(Sqrt[2]*c^(3/2)*Sqrt[b^2 - 4*a*c]*Sqrt[2*c*d
- (b - Sqrt[b^2 - 4*a*c])*e]) + ((2*c^2*d^2 + b*(b + Sqrt[b^2 - 4*a*c])*e^2 - 2
*c*e*(b*d + Sqrt[b^2 - 4*a*c]*d + a*e))*ArcTanh[(Sqrt[2]*Sqrt[c]*Sqrt[d + e*x^2]
)/Sqrt[2*c*d - (b + Sqrt[b^2 - 4*a*c])*e]])/(Sqrt[2]*c^(3/2)*Sqrt[b^2 - 4*a*c]*S
qrt[2*c*d - (b + Sqrt[b^2 - 4*a*c])*e])

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Rubi [A]  time = 3.19187, antiderivative size = 327, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.185 \[ -\frac{\left (-2 c e \left (-d \sqrt{b^2-4 a c}+a e+b d\right )+b e^2 \left (b-\sqrt{b^2-4 a c}\right )+2 c^2 d^2\right ) \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{c} \sqrt{d+e x^2}}{\sqrt{2 c d-e \left (b-\sqrt{b^2-4 a c}\right )}}\right )}{\sqrt{2} c^{3/2} \sqrt{b^2-4 a c} \sqrt{2 c d-e \left (b-\sqrt{b^2-4 a c}\right )}}+\frac{\left (-2 c e \left (d \sqrt{b^2-4 a c}+a e+b d\right )+b e^2 \left (\sqrt{b^2-4 a c}+b\right )+2 c^2 d^2\right ) \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{c} \sqrt{d+e x^2}}{\sqrt{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}}\right )}{\sqrt{2} c^{3/2} \sqrt{b^2-4 a c} \sqrt{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}}+\frac{e \sqrt{d+e x^2}}{c} \]

Antiderivative was successfully verified.

[In]  Int[(x*(d + e*x^2)^(3/2))/(a + b*x^2 + c*x^4),x]

[Out]

(e*Sqrt[d + e*x^2])/c - ((2*c^2*d^2 + b*(b - Sqrt[b^2 - 4*a*c])*e^2 - 2*c*e*(b*d
- Sqrt[b^2 - 4*a*c]*d + a*e))*ArcTanh[(Sqrt[2]*Sqrt[c]*Sqrt[d + e*x^2])/Sqrt[2*
c*d - (b - Sqrt[b^2 - 4*a*c])*e]])/(Sqrt[2]*c^(3/2)*Sqrt[b^2 - 4*a*c]*Sqrt[2*c*d
- (b - Sqrt[b^2 - 4*a*c])*e]) + ((2*c^2*d^2 + b*(b + Sqrt[b^2 - 4*a*c])*e^2 - 2
*c*e*(b*d + Sqrt[b^2 - 4*a*c]*d + a*e))*ArcTanh[(Sqrt[2]*Sqrt[c]*Sqrt[d + e*x^2]
)/Sqrt[2*c*d - (b + Sqrt[b^2 - 4*a*c])*e]])/(Sqrt[2]*c^(3/2)*Sqrt[b^2 - 4*a*c]*S
qrt[2*c*d - (b + Sqrt[b^2 - 4*a*c])*e])

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Rubi in Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x*(e*x**2+d)**(3/2)/(c*x**4+b*x**2+a),x)

[Out]

Timed out

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Mathematica [A]  time = 0.761663, size = 326, normalized size = 1. \[ \frac{\frac{\sqrt{2} \left (2 c e \left (-d \sqrt{b^2-4 a c}+a e+b d\right )+b e^2 \left (\sqrt{b^2-4 a c}-b\right )-2 c^2 d^2\right ) \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{c} \sqrt{d+e x^2}}{\sqrt{e \sqrt{b^2-4 a c}-b e+2 c d}}\right )}{\sqrt{b^2-4 a c} \sqrt{e \left (\sqrt{b^2-4 a c}-b\right )+2 c d}}+\frac{\sqrt{2} \left (-2 c e \left (d \sqrt{b^2-4 a c}+a e+b d\right )+b e^2 \left (\sqrt{b^2-4 a c}+b\right )+2 c^2 d^2\right ) \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{c} \sqrt{d+e x^2}}{\sqrt{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}}\right )}{\sqrt{b^2-4 a c} \sqrt{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}}+2 \sqrt{c} e \sqrt{d+e x^2}}{2 c^{3/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[(x*(d + e*x^2)^(3/2))/(a + b*x^2 + c*x^4),x]

[Out]

(2*Sqrt[c]*e*Sqrt[d + e*x^2] + (Sqrt[2]*(-2*c^2*d^2 + b*(-b + Sqrt[b^2 - 4*a*c])
*e^2 + 2*c*e*(b*d - Sqrt[b^2 - 4*a*c]*d + a*e))*ArcTanh[(Sqrt[2]*Sqrt[c]*Sqrt[d
+ e*x^2])/Sqrt[2*c*d - b*e + Sqrt[b^2 - 4*a*c]*e]])/(Sqrt[b^2 - 4*a*c]*Sqrt[2*c*
d + (-b + Sqrt[b^2 - 4*a*c])*e]) + (Sqrt[2]*(2*c^2*d^2 + b*(b + Sqrt[b^2 - 4*a*c
])*e^2 - 2*c*e*(b*d + Sqrt[b^2 - 4*a*c]*d + a*e))*ArcTanh[(Sqrt[2]*Sqrt[c]*Sqrt[
d + e*x^2])/Sqrt[2*c*d - (b + Sqrt[b^2 - 4*a*c])*e]])/(Sqrt[b^2 - 4*a*c]*Sqrt[2*
c*d - (b + Sqrt[b^2 - 4*a*c])*e]))/(2*c^(3/2))

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Maple [C]  time = 0.028, size = 279, normalized size = 0.9 \[ -{\frac{x}{2\,c}{e}^{{\frac{3}{2}}}}+{\frac{e}{2\,c}\sqrt{e{x}^{2}+d}}+{\frac{e}{4\,c}\sum _{{\it \_R}={\it RootOf} \left ( c{{\it \_Z}}^{8}+ \left ( 4\,be-4\,cd \right ){{\it \_Z}}^{6}+ \left ( 16\,a{e}^{2}-8\,bde+6\,c{d}^{2} \right ){{\it \_Z}}^{4}+ \left ( 4\,b{d}^{2}e-4\,c{d}^{3} \right ){{\it \_Z}}^{2}+c{d}^{4} \right ) }{\frac{ \left ( -be+2\,cd \right ){{\it \_R}}^{6}+ \left ( -4\,a{e}^{2}+3\,bde-2\,c{d}^{2} \right ){{\it \_R}}^{4}+d \left ( 4\,a{e}^{2}-3\,bde+2\,c{d}^{2} \right ){{\it \_R}}^{2}+b{d}^{3}e-2\,c{d}^{4}}{{{\it \_R}}^{7}c+3\,{{\it \_R}}^{5}be-3\,{{\it \_R}}^{5}cd+8\,{{\it \_R}}^{3}a{e}^{2}-4\,{{\it \_R}}^{3}bde+3\,{{\it \_R}}^{3}c{d}^{2}+{\it \_R}\,b{d}^{2}e-{\it \_R}\,c{d}^{3}}\ln \left ( \sqrt{e{x}^{2}+d}-x\sqrt{e}-{\it \_R} \right ) }}+{\frac{de}{2\,c} \left ( \sqrt{e{x}^{2}+d}-x\sqrt{e} \right ) ^{-1}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x*(e*x^2+d)^(3/2)/(c*x^4+b*x^2+a),x)

[Out]

-1/2*e^(3/2)/c*x+1/2*e*(e*x^2+d)^(1/2)/c+1/4*e/c*sum(((-b*e+2*c*d)*_R^6+(-4*a*e^
2+3*b*d*e-2*c*d^2)*_R^4+d*(4*a*e^2-3*b*d*e+2*c*d^2)*_R^2+b*d^3*e-2*c*d^4)/(_R^7*
c+3*_R^5*b*e-3*_R^5*c*d+8*_R^3*a*e^2-4*_R^3*b*d*e+3*_R^3*c*d^2+_R*b*d^2*e-_R*c*d
^3)*ln((e*x^2+d)^(1/2)-x*e^(1/2)-_R),_R=RootOf(c*_Z^8+(4*b*e-4*c*d)*_Z^6+(16*a*e
^2-8*b*d*e+6*c*d^2)*_Z^4+(4*b*d^2*e-4*c*d^3)*_Z^2+c*d^4))+1/2*e/c*d/((e*x^2+d)^(
1/2)-x*e^(1/2))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (e x^{2} + d\right )}^{\frac{3}{2}} x}{c x^{4} + b x^{2} + a}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x^2 + d)^(3/2)*x/(c*x^4 + b*x^2 + a),x, algorithm="maxima")

[Out]

integrate((e*x^2 + d)^(3/2)*x/(c*x^4 + b*x^2 + a), x)

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Fricas [A]  time = 129.552, size = 5999, normalized size = 18.35 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x^2 + d)^(3/2)*x/(c*x^4 + b*x^2 + a),x, algorithm="fricas")

[Out]

-1/4*(sqrt(1/2)*c*sqrt((2*c^3*d^3 - 3*b*c^2*d^2*e + 3*(b^2*c - 2*a*c^2)*d*e^2 -
(b^3 - 3*a*b*c)*e^3 + (b^2*c^3 - 4*a*c^4)*sqrt((9*c^4*d^4*e^2 - 18*b*c^3*d^3*e^3
+ 3*(5*b^2*c^2 - 2*a*c^3)*d^2*e^4 - 6*(b^3*c - a*b*c^2)*d*e^5 + (b^4 - 2*a*b^2*
c + a^2*c^2)*e^6)/(b^2*c^6 - 4*a*c^7)))/(b^2*c^3 - 4*a*c^4))*log(-(6*b*c^3*d^5*e
- 6*(2*b^2*c^2 + a*c^3)*d^4*e^2 + 8*(b^3*c + 2*a*b*c^2)*d^3*e^3 - 2*(b^4 + 6*a*
b^2*c + 2*a^2*c^2)*d^2*e^4 + 2*(2*a*b^3 + a^2*b*c)*d*e^5 - 2*(a^2*b^2 - a^3*c)*e
^6 + (3*b*c^3*d^4*e^2 - 6*b^2*c^2*d^3*e^3 + 2*(2*b^3*c + a*b*c^2)*d^2*e^4 - (b^4
+ 2*a*b^2*c)*d*e^5 + (a*b^3 - a^2*b*c)*e^6)*x^2 + 2*sqrt(1/2)*(3*(b^2*c^3 - 4*a
*c^4)*d^3*e - 6*(b^3*c^2 - 4*a*b*c^3)*d^2*e^2 + (4*b^4*c - 17*a*b^2*c^2 + 4*a^2*
c^3)*d*e^3 - (b^5 - 5*a*b^3*c + 4*a^2*b*c^2)*e^4 + ((b^3*c^4 - 4*a*b*c^5)*d - (b
^4*c^3 - 6*a*b^2*c^4 + 8*a^2*c^5)*e)*sqrt((9*c^4*d^4*e^2 - 18*b*c^3*d^3*e^3 + 3*
(5*b^2*c^2 - 2*a*c^3)*d^2*e^4 - 6*(b^3*c - a*b*c^2)*d*e^5 + (b^4 - 2*a*b^2*c + a
^2*c^2)*e^6)/(b^2*c^6 - 4*a*c^7)))*sqrt(e*x^2 + d)*sqrt((2*c^3*d^3 - 3*b*c^2*d^2
*e + 3*(b^2*c - 2*a*c^2)*d*e^2 - (b^3 - 3*a*b*c)*e^3 + (b^2*c^3 - 4*a*c^4)*sqrt(
(9*c^4*d^4*e^2 - 18*b*c^3*d^3*e^3 + 3*(5*b^2*c^2 - 2*a*c^3)*d^2*e^4 - 6*(b^3*c -
a*b*c^2)*d*e^5 + (b^4 - 2*a*b^2*c + a^2*c^2)*e^6)/(b^2*c^6 - 4*a*c^7)))/(b^2*c^
3 - 4*a*c^4)) + (2*(b^2*c^4 - 4*a*c^5)*d^3 - 2*(b^3*c^3 - 4*a*b*c^4)*d^2*e + 2*(
a*b^2*c^3 - 4*a^2*c^4)*d*e^2 + ((b^2*c^4 - 4*a*c^5)*d^2*e - (b^3*c^3 - 4*a*b*c^4
)*d*e^2 + (a*b^2*c^3 - 4*a^2*c^4)*e^3)*x^2)*sqrt((9*c^4*d^4*e^2 - 18*b*c^3*d^3*e
^3 + 3*(5*b^2*c^2 - 2*a*c^3)*d^2*e^4 - 6*(b^3*c - a*b*c^2)*d*e^5 + (b^4 - 2*a*b^
2*c + a^2*c^2)*e^6)/(b^2*c^6 - 4*a*c^7)))/x^2) - sqrt(1/2)*c*sqrt((2*c^3*d^3 - 3
*b*c^2*d^2*e + 3*(b^2*c - 2*a*c^2)*d*e^2 - (b^3 - 3*a*b*c)*e^3 + (b^2*c^3 - 4*a*
c^4)*sqrt((9*c^4*d^4*e^2 - 18*b*c^3*d^3*e^3 + 3*(5*b^2*c^2 - 2*a*c^3)*d^2*e^4 -
6*(b^3*c - a*b*c^2)*d*e^5 + (b^4 - 2*a*b^2*c + a^2*c^2)*e^6)/(b^2*c^6 - 4*a*c^7)
))/(b^2*c^3 - 4*a*c^4))*log(-(6*b*c^3*d^5*e - 6*(2*b^2*c^2 + a*c^3)*d^4*e^2 + 8*
(b^3*c + 2*a*b*c^2)*d^3*e^3 - 2*(b^4 + 6*a*b^2*c + 2*a^2*c^2)*d^2*e^4 + 2*(2*a*b
^3 + a^2*b*c)*d*e^5 - 2*(a^2*b^2 - a^3*c)*e^6 + (3*b*c^3*d^4*e^2 - 6*b^2*c^2*d^3
*e^3 + 2*(2*b^3*c + a*b*c^2)*d^2*e^4 - (b^4 + 2*a*b^2*c)*d*e^5 + (a*b^3 - a^2*b*
c)*e^6)*x^2 - 2*sqrt(1/2)*(3*(b^2*c^3 - 4*a*c^4)*d^3*e - 6*(b^3*c^2 - 4*a*b*c^3)
*d^2*e^2 + (4*b^4*c - 17*a*b^2*c^2 + 4*a^2*c^3)*d*e^3 - (b^5 - 5*a*b^3*c + 4*a^2
*b*c^2)*e^4 + ((b^3*c^4 - 4*a*b*c^5)*d - (b^4*c^3 - 6*a*b^2*c^4 + 8*a^2*c^5)*e)*
sqrt((9*c^4*d^4*e^2 - 18*b*c^3*d^3*e^3 + 3*(5*b^2*c^2 - 2*a*c^3)*d^2*e^4 - 6*(b^
3*c - a*b*c^2)*d*e^5 + (b^4 - 2*a*b^2*c + a^2*c^2)*e^6)/(b^2*c^6 - 4*a*c^7)))*sq
rt(e*x^2 + d)*sqrt((2*c^3*d^3 - 3*b*c^2*d^2*e + 3*(b^2*c - 2*a*c^2)*d*e^2 - (b^3
- 3*a*b*c)*e^3 + (b^2*c^3 - 4*a*c^4)*sqrt((9*c^4*d^4*e^2 - 18*b*c^3*d^3*e^3 + 3
*(5*b^2*c^2 - 2*a*c^3)*d^2*e^4 - 6*(b^3*c - a*b*c^2)*d*e^5 + (b^4 - 2*a*b^2*c +
a^2*c^2)*e^6)/(b^2*c^6 - 4*a*c^7)))/(b^2*c^3 - 4*a*c^4)) + (2*(b^2*c^4 - 4*a*c^5
)*d^3 - 2*(b^3*c^3 - 4*a*b*c^4)*d^2*e + 2*(a*b^2*c^3 - 4*a^2*c^4)*d*e^2 + ((b^2*
c^4 - 4*a*c^5)*d^2*e - (b^3*c^3 - 4*a*b*c^4)*d*e^2 + (a*b^2*c^3 - 4*a^2*c^4)*e^3
)*x^2)*sqrt((9*c^4*d^4*e^2 - 18*b*c^3*d^3*e^3 + 3*(5*b^2*c^2 - 2*a*c^3)*d^2*e^4
- 6*(b^3*c - a*b*c^2)*d*e^5 + (b^4 - 2*a*b^2*c + a^2*c^2)*e^6)/(b^2*c^6 - 4*a*c^
7)))/x^2) + sqrt(1/2)*c*sqrt((2*c^3*d^3 - 3*b*c^2*d^2*e + 3*(b^2*c - 2*a*c^2)*d*
e^2 - (b^3 - 3*a*b*c)*e^3 - (b^2*c^3 - 4*a*c^4)*sqrt((9*c^4*d^4*e^2 - 18*b*c^3*d
^3*e^3 + 3*(5*b^2*c^2 - 2*a*c^3)*d^2*e^4 - 6*(b^3*c - a*b*c^2)*d*e^5 + (b^4 - 2*
a*b^2*c + a^2*c^2)*e^6)/(b^2*c^6 - 4*a*c^7)))/(b^2*c^3 - 4*a*c^4))*log(-(6*b*c^3
*d^5*e - 6*(2*b^2*c^2 + a*c^3)*d^4*e^2 + 8*(b^3*c + 2*a*b*c^2)*d^3*e^3 - 2*(b^4
+ 6*a*b^2*c + 2*a^2*c^2)*d^2*e^4 + 2*(2*a*b^3 + a^2*b*c)*d*e^5 - 2*(a^2*b^2 - a^
3*c)*e^6 + (3*b*c^3*d^4*e^2 - 6*b^2*c^2*d^3*e^3 + 2*(2*b^3*c + a*b*c^2)*d^2*e^4
- (b^4 + 2*a*b^2*c)*d*e^5 + (a*b^3 - a^2*b*c)*e^6)*x^2 + 2*sqrt(1/2)*(3*(b^2*c^3
- 4*a*c^4)*d^3*e - 6*(b^3*c^2 - 4*a*b*c^3)*d^2*e^2 + (4*b^4*c - 17*a*b^2*c^2 +
4*a^2*c^3)*d*e^3 - (b^5 - 5*a*b^3*c + 4*a^2*b*c^2)*e^4 - ((b^3*c^4 - 4*a*b*c^5)*
d - (b^4*c^3 - 6*a*b^2*c^4 + 8*a^2*c^5)*e)*sqrt((9*c^4*d^4*e^2 - 18*b*c^3*d^3*e^
3 + 3*(5*b^2*c^2 - 2*a*c^3)*d^2*e^4 - 6*(b^3*c - a*b*c^2)*d*e^5 + (b^4 - 2*a*b^2
*c + a^2*c^2)*e^6)/(b^2*c^6 - 4*a*c^7)))*sqrt(e*x^2 + d)*sqrt((2*c^3*d^3 - 3*b*c
^2*d^2*e + 3*(b^2*c - 2*a*c^2)*d*e^2 - (b^3 - 3*a*b*c)*e^3 - (b^2*c^3 - 4*a*c^4)
*sqrt((9*c^4*d^4*e^2 - 18*b*c^3*d^3*e^3 + 3*(5*b^2*c^2 - 2*a*c^3)*d^2*e^4 - 6*(b
^3*c - a*b*c^2)*d*e^5 + (b^4 - 2*a*b^2*c + a^2*c^2)*e^6)/(b^2*c^6 - 4*a*c^7)))/(
b^2*c^3 - 4*a*c^4)) - (2*(b^2*c^4 - 4*a*c^5)*d^3 - 2*(b^3*c^3 - 4*a*b*c^4)*d^2*e
+ 2*(a*b^2*c^3 - 4*a^2*c^4)*d*e^2 + ((b^2*c^4 - 4*a*c^5)*d^2*e - (b^3*c^3 - 4*a
*b*c^4)*d*e^2 + (a*b^2*c^3 - 4*a^2*c^4)*e^3)*x^2)*sqrt((9*c^4*d^4*e^2 - 18*b*c^3
*d^3*e^3 + 3*(5*b^2*c^2 - 2*a*c^3)*d^2*e^4 - 6*(b^3*c - a*b*c^2)*d*e^5 + (b^4 -
2*a*b^2*c + a^2*c^2)*e^6)/(b^2*c^6 - 4*a*c^7)))/x^2) - sqrt(1/2)*c*sqrt((2*c^3*d
^3 - 3*b*c^2*d^2*e + 3*(b^2*c - 2*a*c^2)*d*e^2 - (b^3 - 3*a*b*c)*e^3 - (b^2*c^3
- 4*a*c^4)*sqrt((9*c^4*d^4*e^2 - 18*b*c^3*d^3*e^3 + 3*(5*b^2*c^2 - 2*a*c^3)*d^2*
e^4 - 6*(b^3*c - a*b*c^2)*d*e^5 + (b^4 - 2*a*b^2*c + a^2*c^2)*e^6)/(b^2*c^6 - 4*
a*c^7)))/(b^2*c^3 - 4*a*c^4))*log(-(6*b*c^3*d^5*e - 6*(2*b^2*c^2 + a*c^3)*d^4*e^
2 + 8*(b^3*c + 2*a*b*c^2)*d^3*e^3 - 2*(b^4 + 6*a*b^2*c + 2*a^2*c^2)*d^2*e^4 + 2*
(2*a*b^3 + a^2*b*c)*d*e^5 - 2*(a^2*b^2 - a^3*c)*e^6 + (3*b*c^3*d^4*e^2 - 6*b^2*c
^2*d^3*e^3 + 2*(2*b^3*c + a*b*c^2)*d^2*e^4 - (b^4 + 2*a*b^2*c)*d*e^5 + (a*b^3 -
a^2*b*c)*e^6)*x^2 - 2*sqrt(1/2)*(3*(b^2*c^3 - 4*a*c^4)*d^3*e - 6*(b^3*c^2 - 4*a*
b*c^3)*d^2*e^2 + (4*b^4*c - 17*a*b^2*c^2 + 4*a^2*c^3)*d*e^3 - (b^5 - 5*a*b^3*c +
4*a^2*b*c^2)*e^4 - ((b^3*c^4 - 4*a*b*c^5)*d - (b^4*c^3 - 6*a*b^2*c^4 + 8*a^2*c^
5)*e)*sqrt((9*c^4*d^4*e^2 - 18*b*c^3*d^3*e^3 + 3*(5*b^2*c^2 - 2*a*c^3)*d^2*e^4 -
6*(b^3*c - a*b*c^2)*d*e^5 + (b^4 - 2*a*b^2*c + a^2*c^2)*e^6)/(b^2*c^6 - 4*a*c^7
)))*sqrt(e*x^2 + d)*sqrt((2*c^3*d^3 - 3*b*c^2*d^2*e + 3*(b^2*c - 2*a*c^2)*d*e^2
- (b^3 - 3*a*b*c)*e^3 - (b^2*c^3 - 4*a*c^4)*sqrt((9*c^4*d^4*e^2 - 18*b*c^3*d^3*e
^3 + 3*(5*b^2*c^2 - 2*a*c^3)*d^2*e^4 - 6*(b^3*c - a*b*c^2)*d*e^5 + (b^4 - 2*a*b^
2*c + a^2*c^2)*e^6)/(b^2*c^6 - 4*a*c^7)))/(b^2*c^3 - 4*a*c^4)) - (2*(b^2*c^4 - 4
*a*c^5)*d^3 - 2*(b^3*c^3 - 4*a*b*c^4)*d^2*e + 2*(a*b^2*c^3 - 4*a^2*c^4)*d*e^2 +
((b^2*c^4 - 4*a*c^5)*d^2*e - (b^3*c^3 - 4*a*b*c^4)*d*e^2 + (a*b^2*c^3 - 4*a^2*c^
4)*e^3)*x^2)*sqrt((9*c^4*d^4*e^2 - 18*b*c^3*d^3*e^3 + 3*(5*b^2*c^2 - 2*a*c^3)*d^
2*e^4 - 6*(b^3*c - a*b*c^2)*d*e^5 + (b^4 - 2*a*b^2*c + a^2*c^2)*e^6)/(b^2*c^6 -
4*a*c^7)))/x^2) - 4*sqrt(e*x^2 + d)*e)/c

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x \left (d + e x^{2}\right )^{\frac{3}{2}}}{a + b x^{2} + c x^{4}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x*(e*x**2+d)**(3/2)/(c*x**4+b*x**2+a),x)

[Out]

Integral(x*(d + e*x**2)**(3/2)/(a + b*x**2 + c*x**4), x)

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GIAC/XCAS [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x^2 + d)^(3/2)*x/(c*x^4 + b*x^2 + a),x, algorithm="giac")

[Out]

Timed out